About this Journal Submit a Manuscript Table of Contents
Journal of Function Spaces and Applications
VolumeΒ 2012Β (2012), Article IDΒ 765903, 18 pages
http://dx.doi.org/10.1155/2012/765903
Research Article

π‘ž-Hyperconvexity in Quasipseudometric Spaces and Fixed Point Theorems

Department of Mathematics and Applied Mathematics, University of Cape Town, Rondebosch 7701, South Africa

Received 15 May 2012; Accepted 4 July 2012

Academic Editor: SalvadorΒ Romaguera

Copyright Β© 2012 Hans-Peter A. KΓΌnzi and Olivier Olela Otafudu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In a previous work, we started investigating the concept of hyperconvexity in quasipseudometric spaces which we called π‘ž-hyperconvexity or Isbell-convexity. In this paper, we continue our studies of this concept, generalizing further known results about hyperconvexity from the metric setting to our theory. In particular, in the present paper, we consider subspaces of q-hyperconvex spaces and also present some fixed point theorems for nonexpansive self-maps on a bounded q-hyperconvex quasipseudometric space. In analogy with a metric result, we show among other things that a set-valued mapping π‘‡βˆ— on a q-hyperconvex 𝑇0-quasimetric space (X, d) which takes values in the space of nonempty externally q-hyperconvex subsets of (X, d) always has a single-valued selection T which satisfies 𝑑(𝑇(π‘₯),𝑇(𝑦))≀𝑑𝐻(π‘‡βˆ—(π‘₯),π‘‡βˆ—(𝑦)) whenever π‘₯,π‘¦βˆˆπ‘‹. (Here, 𝑑𝐻 denotes the usual (extended) Hausdorff quasipseudometric determined by 𝑑 on the set 𝒫0(𝑋) of nonempty subsets of X.)

1. Introduction

In a previous work, we started investigating a concept of hyperconvexity in quasipseudometric spaces, which we called π‘ž-hyperconvexity or Isbell-convexity (see [1], compare [2]). In this paper, we continue our studies of this concept by generalizing further known results about hyperconvexity from the metric setting to our theory. Among other things, in the present paper we consider subspaces of π‘ž-hyperconvex spaces and also present some fixed point theorems. In particular, we show that a set-valued mapping π‘‡βˆ— on a π‘ž-hyperconvex 𝑇0-quasimetric space (𝑋,𝑑) which takes values in the space of nonempty externally π‘ž-hyperconvex subsets of (𝑋,𝑑) always has a single-valued selection 𝑇 which satisfies 𝑑(𝑇(π‘₯),𝑇(𝑦))≀𝑑𝐻(π‘‡βˆ—(π‘₯),π‘‡βˆ—(𝑦)) whenever π‘₯,π‘¦βˆˆπ‘‹. (Here, 𝑑𝐻 denotes the usual (extended) Hausdorff quasipseudometric determined by 𝑑 on the set 𝒫0(𝑋) of nonempty subsets of 𝑋.)

Our investigations confirm the surprising fact that many classical results about hyperconvexity in metric spaces do not make essential use of the symmetry of the metric and, therefore, still holdβ€”in a sometimes slightly modified formβ€”for our concept of π‘ž-hyperconvexity in quasipseudometric spaces (see also [3] for a more general approach).

For the basic facts concerning quasipseudometrics and quasiuniformities we refer the reader to [4, 5]. Some recent work about quasipseudometric spaces can be found in the articles [6–9].

2. Preliminaries

In order to fix the terminology, we start with some basic concepts.

Definition 2.1. Let 𝑋 be a set and let π‘‘βˆΆπ‘‹Γ—π‘‹β†’[0,∞) be a function mapping into the set [0,∞) of the nonnegative reals. Then, 𝑑 is called a quasipseudometric on 𝑋 if(a)𝑑(π‘₯,π‘₯)=0 whenever π‘₯βˆˆπ‘‹,(b)𝑑(π‘₯,𝑧)≀𝑑(π‘₯,𝑦)+𝑑(𝑦,𝑧) whenever π‘₯,𝑦,π‘§βˆˆπ‘‹.
We will say that 𝑑 is a 𝑇0-quasimetric provided that 𝑑 also satisfies the following condition: for each π‘₯,π‘¦βˆˆπ‘‹, 𝑑(π‘₯,𝑦)=0=𝑑(𝑦,π‘₯)impliesthatπ‘₯=𝑦.(2.1)

Remark 2.2. Let 𝑑 be a quasipseudometric on a set 𝑋, then π‘‘βˆ’1βˆΆπ‘‹Γ—π‘‹β†’[0,∞) defined by π‘‘βˆ’1(π‘₯,𝑦)=𝑑(𝑦,π‘₯) whenever π‘₯,π‘¦βˆˆπ‘‹ is also a quasipseudometric, called the conjugate quasipseudometric of 𝑑. As usual, a quasipseudometric 𝑑 on 𝑋 such that 𝑑=π‘‘βˆ’1 is called a pseudometric. Note that for any (𝑇0)-quasipseudometric 𝑑, the function 𝑑𝑠=max{𝑑,π‘‘βˆ’1}=π‘‘βˆ¨π‘‘βˆ’1 is a pseudometric (metric).
For any π‘Ž,π‘βˆˆ[0,∞), we will set π‘ŽΜ‡βˆ’π‘=max{π‘Žβˆ’π‘,0}.
Let (𝑋,𝑑) be a quasipseudometric space. For each π‘₯βˆˆπ‘‹ and πœ–>0, the set 𝐡𝑑(π‘₯,πœ–)={π‘¦βˆˆπ‘‹βˆΆπ‘‘(π‘₯,𝑦)<πœ–} denotes the open πœ–-ball at π‘₯. The collection of all β€œopen” balls yields a base for a topology 𝜏(𝑑). It is called the topology induced by 𝑑 on 𝑋. Similarly, for each π‘₯βˆˆπ‘‹ and πœ–β‰₯0, we define the ball 𝐢𝑑(π‘₯,πœ–)={π‘¦βˆˆπ‘‹βˆΆπ‘‘(π‘₯,𝑦)β‰€πœ–}. Note that this latter set is 𝜏(π‘‘βˆ’1)-closed, but not 𝜏(𝑑)-closed in general. As usual, in the theory of quasiuniformities, for a subset 𝐴 of 𝑋 and πœ–>0, we will also use notations like 𝐡𝑑⋃(𝐴,πœ–)=π‘Žβˆˆπ΄π΅π‘‘(π‘Ž,πœ–) and similarly 𝐢𝑑⋃(𝐴,πœ–)=π‘Žβˆˆπ΄πΆπ‘‘(π‘Ž,πœ–).
A pair (𝐢𝑑(π‘₯,π‘Ÿ);πΆπ‘‘βˆ’1(π‘₯,𝑠)) with π‘₯βˆˆπ‘‹ and nonnegative reals π‘Ÿ,𝑠 will be called a double ball at π‘₯.
We shall also speak of a family [(𝐢𝑑(π‘₯𝑖,π‘Ÿπ‘–))π‘–βˆˆπΌ;(πΆπ‘‘βˆ’1(π‘₯𝑖,𝑠𝑖))π‘–βˆˆπΌ] of double balls, with π‘₯π‘–βˆˆπ‘‹ and π‘Ÿπ‘–,𝑠𝑖β‰₯0 whenever π‘–βˆˆπΌ.
Let (𝑋,𝑑) be a quasipseudometric space and let 𝒫0(𝑋) be the set of all nonempty subsets of 𝑋. Given πΆβˆˆπ’«0(𝑋), we will set dist(π‘₯,𝐢)=inf{𝑑(π‘₯,𝑐)βˆΆπ‘βˆˆπΆ} and dist(𝐢,π‘₯)=inf{𝑑(𝑐,π‘₯)βˆΆπ‘βˆˆπΆ} whenever π‘₯βˆˆπ‘‹.
For any 𝐴,π΅βˆˆπ’«0(𝑋), we set 𝑑𝐻(𝐴,𝐡)=maxsupπ‘βˆˆπ΅dist(𝐴,𝑏),supπ‘Žβˆˆπ΄ξ‚Όdist(π‘Ž,𝐡)(2.2) (compare [10]).
Then 𝑑𝐻, is the so-called extended (as usual, a quasipseudometric that maps into [0,∞] (instead of [0,∞)) will be called extended)  Hausdorff(-Bourbaki) quasipseudometric on 𝒫0(𝑋). It is known that 𝑑𝐻 is an extended 𝑇0-quasimetric when restricted to the set of all the nonempty subsets 𝐴 of 𝑋 which satisfy 𝐴=cl𝜏(𝑑)𝐴∩cl𝜏(π‘‘βˆ’1)𝐴 (compare [11, page 164]).
A map π‘“βˆΆ(𝑋,𝑑)β†’(π‘Œ,𝑒) between two quasipseudometric spaces (𝑋,𝑑) and (π‘Œ,𝑒) is called an isometry or isometric map provided that 𝑒(𝑓(π‘₯),𝑓(𝑦))=𝑑(π‘₯,𝑦) whenever π‘₯,π‘¦βˆˆπ‘‹. Two quasipseudometric spaces (𝑋,𝑑) and (π‘Œ,𝑒) will be called isometric provided that there exists a bijective isometry π‘“βˆΆ(𝑋,𝑑)β†’(π‘Œ,𝑒). A map π‘“βˆΆ(𝑋,𝑑)β†’(π‘Œ,𝑒) between two quasipseudometric spaces (𝑋,𝑑) and (π‘Œ,𝑒) is called nonexpansive provided that 𝑒(𝑓(π‘₯),𝑓(𝑦))≀𝑑(π‘₯,𝑦) whenever π‘₯,π‘¦βˆˆπ‘‹.
The following definitions can be found in [1] (compare [12]).

Definition 2.3 (see [1, Definition 2]). A quasipseudometric space (𝑋,𝑑) is called π‘ž-hyperconvex (or Isbell-convex) provided that for each family (π‘₯𝑖)π‘–βˆˆπΌ of points in 𝑋 and families (π‘Ÿπ‘–)π‘–βˆˆπΌ and (𝑠𝑖)π‘–βˆˆπΌ of nonnegative real numbers satisfying 𝑑(π‘₯𝑖,π‘₯𝑗)β‰€π‘Ÿπ‘–+𝑠𝑗 whenever 𝑖,π‘—βˆˆπΌ, the following condition holds: ξ™π‘–βˆˆπΌξ€·πΆπ‘‘ξ€·π‘₯𝑖,π‘Ÿπ‘–ξ€Έβˆ©πΆπ‘‘βˆ’1ξ€·π‘₯𝑖,π‘ π‘–ξ€Έξ€Έβ‰ βˆ….(2.3)

Definition 2.4 (see [1, Definition 5]). Let (𝑋,𝑑) be a quasipseudometric space. A family of double balls [(𝐢𝑑(π‘₯𝑖,π‘Ÿπ‘–))π‘–βˆˆπΌ;(πΆπ‘‘βˆ’1(π‘₯𝑖,𝑠𝑖))π‘–βˆˆπΌ] with π‘Ÿπ‘–,π‘ π‘–βˆˆ[0,∞) and π‘₯π‘–βˆˆπ‘‹ whenever π‘–βˆˆπΌ is said to have the mixed binary intersection property if for all indices 𝑖,π‘—βˆˆπΌ,𝐢𝑑(π‘₯𝑖,π‘Ÿπ‘–)βˆ©πΆπ‘‘βˆ’1(π‘₯𝑗,𝑠𝑗)β‰ βˆ….

Definition 2.5 (see [1, Definition 6]). A quasipseudometric space (𝑋,𝑑) is called π‘ž-hypercomplete (or Isbell-complete) if every family 𝐢𝑑π‘₯𝑖,π‘Ÿπ‘–ξ€Έξ€Έπ‘–βˆˆπΌ;ξ€·πΆπ‘‘βˆ’1ξ€·π‘₯𝑖,π‘ π‘–ξ€Έξ€Έπ‘–βˆˆπΌξ€»(2.4) of double balls, where π‘Ÿπ‘–,𝑠𝑖β‰₯0 and π‘₯π‘–βˆˆπ‘‹ whenever π‘–βˆˆπΌ, having the mixed binary intersection property satisfies β‹‚π‘–βˆˆπΌ(𝐢𝑑(π‘₯𝑖,π‘Ÿπ‘–)βˆ©πΆπ‘‘βˆ’1(π‘₯𝑖,𝑠𝑖))β‰ βˆ….

Definition 2.6 (see [1, Definition 4]). Let (𝑋,𝑑) be a quasipseudometric space. We say that 𝑋 is metrically convex if for any points π‘₯,π‘¦βˆˆπ‘‹ and nonnegative real numbers π‘Ÿ and 𝑠 such that 𝑑(π‘₯,𝑦)β‰€π‘Ÿ+𝑠, there exists π‘§βˆˆπ‘‹ such that 𝑑(π‘₯,𝑧)β‰€π‘Ÿ and 𝑑(𝑧,𝑦)≀𝑠.
The following useful result was established in [1, Proposition 1]. A quasipseudometric space (𝑋,𝑑) is π‘ž-hyperconvex if and only if it is metrically convex and π‘ž-hypercomplete.
As usual, a subset 𝐴 of a quasipseudometric space (𝑋,𝑑) will be called bounded provided that there is a positive real constant 𝑀 such that 𝑑(π‘₯,𝑦)<𝑀 whenever π‘₯,π‘¦βˆˆπ΄. Note that a subset 𝐴 of (𝑋,𝑑) is bounded if and only if there are π‘₯βˆˆπ‘‹ and π‘Ÿ,𝑠β‰₯0 such that π΄βŠ†πΆπ‘‘(π‘₯,π‘Ÿ)βˆ©πΆπ‘‘βˆ’1(π‘₯,𝑠).

3. Some First Results

Proposition 3.1 (compare [13, Proposition 4.5]). Let (𝑋,𝑑) be a π‘ž-hyperconvex quasipseudometric space. Let (π‘₯𝑖)π‘–βˆˆπΌ be a nonempty family of points in 𝑋 and let (π‘Ÿπ‘–)π‘–βˆˆπΌ and (𝑠𝑖)π‘–βˆˆπΌ be two families of nonnegative reals such that 𝑑(π‘₯𝑖,π‘₯𝑗)β‰€π‘Ÿπ‘–+𝑠𝑗. Set ⋂𝐷=π‘–βˆˆπΌ(𝐢𝑑(π‘₯𝑖,π‘Ÿπ‘–)βˆ©πΆπ‘‘βˆ’1(π‘₯𝑖,𝑠𝑖)). Then 𝐷 is nonempty and π‘ž-hyperconvex.

Proof. Note first that π·β‰ βˆ… by π‘ž-hyperconvexity of 𝑋. For each π›Όβˆˆπ‘†, let π‘₯π›Όβˆˆπ· and let π‘Ÿπ›Ό,𝑠𝛼 be nonnegative reals such that 𝑑(π‘₯𝛼,π‘₯𝛽)β‰€π‘Ÿπ›Ό+𝑠𝛽 whenever 𝛼,π›½βˆˆπ‘†.
We show that the family 𝐢𝑑π‘₯𝛼,π‘Ÿπ›Όξ€Έξ€Έπ›Όβˆˆπ‘†,𝐢𝑑π‘₯𝑖,π‘Ÿπ‘–ξ€Έξ€Έπ‘–βˆˆπΌ;ξ€·πΆπ‘‘βˆ’1ξ€·π‘₯𝛼,π‘ π›Όξ€Έξ€Έπ›Όβˆˆπ‘†,ξ€·πΆπ‘‘βˆ’1ξ€·π‘₯𝑖,π‘ π‘–ξ€Έξ€Έπ‘–βˆˆπΌξ€»(3.1) satisfies the hypothesis of π‘ž-hyperconvexity. Indeed, in particular, for each π›Όβˆˆπ‘† and π‘–βˆˆπΌ, we have that 𝑑(π‘₯𝛼,π‘₯𝑖)β‰€π‘ π‘–β‰€π‘Ÿπ›Ό+𝑠𝑖 and 𝑑(π‘₯𝑖,π‘₯𝛼)β‰€π‘Ÿπ‘–β‰€π‘Ÿπ‘–+𝑠𝛼.
Hence, by π‘ž-hyperconvexity of 𝑋, we have that ξ™βˆ…β‰ π‘–βˆˆπΌξ€·πΆπ‘‘ξ€·π‘₯𝑖,π‘Ÿπ‘–ξ€Έβˆ©πΆπ‘‘βˆ’1ξ€·π‘₯𝑖,π‘ π‘–βˆ©ξ™ξ€Έξ€Έπ›Όβˆˆπ‘†ξ€·πΆπ‘‘ξ€·π‘₯𝛼,π‘Ÿπ›Όξ€Έβˆ©πΆπ‘‘βˆ’1ξ€·π‘₯𝛼,𝑠𝛼=π·βˆ©π›Όβˆˆπ‘†ξ€·πΆπ‘‘ξ€·π‘₯𝛼,π‘Ÿπ›Όξ€Έβˆ©πΆπ‘‘βˆ’1ξ€·π‘₯𝛼,𝑠𝛼.ξ€Έξ€Έ(3.2) Hence, the subspace 𝐷 of 𝑋 is π‘ž-hyperconvex.

Let (𝑋,𝑑) be a quasipseudometric space. For a nonempty bounded subset 𝐴 of 𝑋, we set bicov(𝐴)+=𝐢𝑑(π‘₯,π‘Ÿ)βˆΆπ΄βŠ†πΆπ‘‘ξ€Ύ,(π‘₯,π‘Ÿ),π‘₯βˆˆπ‘‹,π‘Ÿβ‰₯0bicov(𝐴)βˆ’=ξ™ξ€½πΆπ‘‘βˆ’1(π‘₯,𝑠)βˆΆπ΄βŠ†πΆπ‘‘βˆ’1ξ€Ύ.(π‘₯,𝑠),π‘₯βˆˆπ‘‹,𝑠β‰₯0(3.3) Furthermore, we define the bicover of 𝐴 by bicov(𝐴)∢=bicov(𝐴)+∩bicov(𝐴)βˆ’.

A nonempty bounded set 𝐴 in a quasipseudometric space (𝑋,𝑑) that can be written as the intersection of a nonempty family of sets of the form 𝐢𝑑(π‘₯,πœ–1)βˆ©πΆπ‘‘βˆ’1(π‘₯,πœ–2) where πœ–1,πœ–2β‰₯0 and π‘₯βˆˆπ‘‹, that is, 𝐴=bicov𝐴, will be called π‘ž-admissible in the following. By π’œπ‘ž(𝑋), we will denote the set of π‘ž-admissible subsets of 𝑋. Note that if (𝑋,𝑑) is π‘ž-hyperconvex, then any member of π’œπ‘ž(𝑋) is π‘ž-hyperconvex in the light of Proposition 3.1.

Let (𝑋,𝑑) be a quasipseudometric space and let 𝐴 be a nonempty bounded subset in (𝑋,𝑑). Then, in accordance with [13, page 79], we can define the cover cov𝐴 of 𝐴 as follows: β‹‚cov𝐴={𝐢𝑑𝑠(π‘₯)βˆΆπ΄βŠ†πΆπ‘‘π‘ (π‘₯),π‘₯βˆˆπ‘‹}. Obviously, we have π΄βŠ†bicov(𝐴)βŠ†cov(𝐴). The latter inclusion can be strict, as our first example shows.

Example 3.2. Let 𝑋=[0,1]Γ—[1/4,3/4] be equipped with the 𝑇0-quasimetric 𝑑 defined by 𝑑((𝛼,𝛽),(𝛼′,𝛽′))=(π›ΌΜ‡βˆ’π›Όβ€²)∨(π›½Μ‡βˆ’π›½β€²) whenever (𝛼,𝛽),(𝛼′,𝛽′)βˆˆπ‘‹.
Consider 𝐴={(0,1/2),(1,1/2)}βŠ†π‘‹. Then, bicov(𝐴) is equal to the line segment in 𝑋 from π‘₯=(0,1/2) to 𝑦=(1,1/2). This follows from the fact that, for each πœ–βˆˆ[0,1/4], we have π‘₯∈[0,1]Γ—[1/4,(1/2)+πœ–]=πΆπ‘‘βˆ’1(𝑦,πœ–) and π‘¦βˆˆ[0,1]Γ—[(1/2)βˆ’πœ–,3/4]=𝐢𝑑(π‘₯,πœ–), and that the line segment is a subset of any set of the form 𝐢𝑑(π‘Ž,π‘Ÿ)βˆ©πΆπ‘‘βˆ’1(𝑏,𝑠) for which {π‘₯,𝑦}βŠ†πΆπ‘‘(π‘Ž,π‘Ÿ)βˆ©πΆπ‘‘βˆ’1(𝑏,𝑠). Indeed, assume that the point 𝑧 belongs to this segment. Then 𝑑(𝑧,𝑦)=0=𝑑(π‘₯,𝑧) and, therefore, π‘§βˆˆπΆπ‘‘(π‘Ž,π‘Ÿ)βˆ©πΆπ‘‘βˆ’1(𝑏,𝑠) by the triangle inequality.
On the other hand, cov(𝐴)=𝑋, since {π‘₯,𝑦}βŠ†πΆπ‘‘π‘ (𝑧,πœ–) with π‘§βˆˆπ‘‹ implies that πœ–β‰₯1/2. Indeed, assume that 𝑧=(π‘Ž,𝑏)βˆˆπ‘‹. Then, π‘Žβ‰€π‘‘π‘ ((π‘Ž,𝑏),(0,1/2))β‰€πœ– and 1βˆ’π‘Žβ‰€π‘‘π‘ ((π‘Ž,𝑏),(1,1/2))β‰€πœ–. Thus, πœ–β‰₯max{π‘Ž,1βˆ’π‘Ž}β‰₯1/2 with π‘Žβˆˆ[0,1]. In the light that the interval [1/4,3/4] has length 1/2, it follows that π‘‹βŠ†πΆπ‘‘π‘ (𝑧,πœ–). Therefore, cov(𝐴)=𝑋.
By the results of [1, Example 1], (bicov(𝐴),𝑑) is π‘ž-hyperconvex, while the metric space (bicov(𝐴),𝑑𝑠) is hyperconvex [1, Proposition 2], but not π‘ž-hyperconvex (see [1, Example 2]).
The following result gives a quasipseudometric variant of a well-known result usually attibuted to Sine [14] (compare also [15]).

Theorem 3.3. If (𝑋,𝑑) is a bounded π‘ž-hyperconvex 𝑇0-quasimetric space and ifβ€‰β€‰π‘‡βˆΆ(𝑋,𝑑)β†’(𝑋,𝑑) is a nonexpansive map, then the fixed point set Fix(𝑇) of   𝑇 in (𝑋,𝑑) is nonempty and π‘ž-hyperconvex.

Proof. We first show that Fix(𝑇)β‰ βˆ…. Note that π‘‡βˆΆ(𝑋,𝑑𝑠)β†’(𝑋,𝑑𝑠) is nonexpansive, since for any π‘₯,π‘¦βˆˆπ‘‹, we have 𝑑(𝑇π‘₯,𝑇𝑦)≀𝑑(π‘₯,𝑦) and 𝑑(𝑇𝑦,𝑇π‘₯)≀𝑑(𝑦,π‘₯), and thus 𝑑𝑠(𝑇π‘₯,𝑇𝑦)≀𝑑𝑠(π‘₯,𝑦). By assumption (𝑋,𝑑𝑠) is bounded. Furthermore, (𝑋,𝑑𝑠) is a hyperconvex space according to [1, Proposition 2]. So, by [13, Theorem 4.8], we know that 𝑇 has a fixed point and Fix(𝑇) is hyperconvex in (𝑋,𝑑𝑠).
We need to show that Fix(𝑇) is indeed π‘ž-hyperconvex. Let ξ€Ί(𝐢𝑑(π‘₯𝑖,π‘Ÿπ‘–))π‘–βˆˆπΌ;(πΆπ‘‘βˆ’1(π‘₯𝑖,𝑠𝑖))π‘–βˆˆπΌξ€»(3.4) be a nonempty family of double balls, where π‘₯π‘–βˆˆFix(𝑇) and (π‘Ÿπ‘–)π‘–βˆˆπΌ and (𝑠𝑖)π‘–βˆˆπΌ are two families of nonnegative reals such that 𝑑(π‘₯𝑖,π‘₯𝑗)β‰€π‘Ÿπ‘–+𝑠𝑗 whenever 𝑖,π‘—βˆˆπΌ. Since 𝑋 is a π‘ž-hyperconvex 𝑇0-quasimetric space, the set 𝑋0=ξ™π‘–βˆˆπΌξ€·πΆπ‘‘ξ€·π‘₯𝑖,π‘Ÿπ‘–ξ€Έβˆ©πΆπ‘‘βˆ’1ξ€·π‘₯𝑖,π‘ π‘–ξ€Έξ€Έβ‰ βˆ….(3.5) Let π‘₯βˆˆπ‘‹0. Then, 𝑑(𝑇(π‘₯),π‘₯𝑖)=𝑑(𝑇(π‘₯),𝑇(π‘₯𝑖))≀𝑑(π‘₯,π‘₯𝑖)≀𝑠𝑖 and 𝑑π‘₯𝑖𝑇π‘₯,𝑇(π‘₯)=𝑑𝑖π‘₯,𝑇(π‘₯)≀𝑑𝑖,π‘₯β‰€π‘Ÿπ‘–(3.6) whenever π‘–βˆˆπΌ. Thus, 𝑇(π‘₯)βˆˆπ‘‹0 and we have π‘‡βˆΆπ‘‹0→𝑋0.
Moreover, 𝑋0 is a bounded π‘ž-hyperconvex 𝑇0-quasimetric space by Proposition 3.1. So the first part of the proof implies that 𝑇 has a fixed point in 𝑋0, which implies that Fix(Tβ‹‚)∩[π‘–βˆˆπΌ(𝐢𝑑(π‘₯𝑖,π‘Ÿπ‘–)βˆ©πΆπ‘‘βˆ’1(π‘₯𝑖,𝑠𝑖)]β‰ βˆ…. We have shown that Fix(𝑇) is π‘ž-hyperconvex.

4. Chains of π‘ž-Hyperconvex Subspaces

In this section, we will prove the analogue of a famous theorem due to Baillon [16].

Theorem 4.1. Let (𝑋,𝑑) be a bounded 𝑇0-quasimetric space and let (𝐻𝑖)π‘–βˆˆπΌ be a descending family of nonempty π‘ž-hyperconvex subsets of 𝑋, where one assumes that 𝐼 is totally ordered such that 𝑖1,𝑖2∈𝐼 and 𝑖1≀𝑖2 hold if and only if 𝐻𝑖2βŠ†π»π‘–1. Then, β‹‚π‘–βˆˆπΌπ»π‘– is nonempty and π‘ž-hyperconvex.

Proof. We begin by showing that β‹‚π‘–βˆˆπΌπ»π‘–β‰ βˆ…. We first note that (𝑋,𝑑𝑠) is a bounded metric space and (𝐻𝑖)π‘–βˆˆπΌ is a descending chain of hyperconvex sets in (𝑋,𝑑𝑠) by [1, Proposition 2]. By the well-known result of Baillon [16, Theorem 7], we conclude that β‹‚π‘–βˆˆπΌπ»π‘– is nonempty and hyperconvex in (𝑋,𝑑𝑠).
In order to complete the proof, we need to show that ⋂𝐻=π‘–βˆˆπΌπ»π‘– is π‘ž-hyperconvex. Let a nonempty family (π‘₯𝛼)π›Όβˆˆπ‘† of points in 𝐻 and families of nonnegative real numbers (π‘Ÿπ›Ό)π›Όβˆˆπ‘† and (𝑠𝛼)π›Όβˆˆπ‘† be given such that 𝑑(π‘₯𝛼,π‘₯𝛽)β‰€π‘Ÿπ›Ό+𝑠𝛽 whenever 𝛼,π›½βˆˆπ‘†. Fix π‘–βˆˆπΌ. Since 𝐻𝑖 is a π‘ž-hyperconvex space and since π‘₯π›Όβˆˆπ»π‘– whenever π›Όβˆˆπ‘†, therefore, π’Ÿπ‘–=β‹‚π›Όβˆˆπ‘†(𝐢𝑑(π‘₯𝛼,π‘Ÿπ›Ό)βˆ©πΆπ‘‘βˆ’1(π‘₯𝛼,𝑠𝛼))βˆ©π»π‘– is nonempty and π‘ž-hyperconvex by the proof of Proposition 3.1 and thus a hyperconvex subset of (𝑋,𝑑𝑠) by [1, Proposition 2].
Thus by the first part of our present proof, ξ™βˆ…β‰ π‘–βˆˆπΌπ’Ÿπ‘–=ξ™π‘–βˆˆπΌξƒ¬ξ™π›Όβˆˆπ‘†ξ€·πΆπ‘‘ξ€·π‘₯𝛼,π‘Ÿπ›Όξ€Έβˆ©πΆπ‘‘βˆ’1ξ€·π‘₯𝛼,π‘ π›Όξ€Έξ€Έβˆ©π»π‘–ξƒ­=ξ™π›Όβˆˆπ‘†ξ€·πΆπ‘‘ξ€·π‘₯𝛼,π‘Ÿπ›Όξ€Έβˆ©πΆπ‘‘βˆ’1ξ€·π‘₯𝛼,π‘ π›Όβˆ©ξ™ξ€Έξ€Έπ‘–βˆˆπΌπ»π‘–,(4.1) since (π’Ÿπ‘–)π‘–βˆˆπΌ is descending. This proves that ⋂𝐻=π‘–βˆˆπΌπ»π‘– is π‘ž-hyperconvex.

Definition 4.2. Let (𝑋,𝑑) be a 𝑇0-quasimetric space and let a family of nonexpansive maps (𝑇𝑖)π‘–βˆˆπΌ, with π‘‡π‘–βˆΆ(𝑋,𝑑)β†’(𝑋,𝑑), be given. We say that (𝑇𝑖)π‘–βˆˆπΌ is a commuting family if π‘‡π‘–βˆ˜π‘‡π‘—=π‘‡π‘—βˆ˜π‘‡π‘– whenever 𝑖,π‘—βˆˆπΌ.
Our next lemma is motivated by [16, Corollary 8].

Lemma 4.3. If (𝐻𝛼)π›Όβˆˆπ‘† is a family of bounded π‘ž-hyperconvex subsets of a 𝑇0-quasimetric space 𝑋 such that β‹‚π›ΌβˆˆπΉπ»π›Ό is nonempty and π‘ž-hyperconvex whenever πΉβŠ†π‘† is finite, then the intersection β‹‚π›Όβˆˆπ‘†π»π›Ό is nonempty and π‘ž-hyperconvex.

Proof. Consider β„±={πΌβŠ†π‘†βˆΆ for all 𝐽 finite, π½βŠ†π‘†,   ⋂𝐼βˆͺ𝐽𝐻𝛼 is nonempty and π‘ž-hyperconvex}.
Obviously βˆ…βˆˆβ„± and β„± satisfies the hypothesis of Zorn's lemma because of Theorem 4.1. Let 𝐼 be maximal in β„±. Then, 𝐼βˆͺ{𝛼}βˆˆβ„± whenever π›Όβˆˆπ‘†. Because of the maximality of 𝐼, we, therefore, have π›ΌβˆˆπΌ whenever π›Όβˆˆπ‘†.

The next result is a consequence of Theorems 3.3 and 4.1. It is analogous to [17, Theorem 6.2].

Theorem 4.4. Let (𝑋,𝑑) be a bounded π‘ž-hyperconvex 𝑇0-quasimetric space. Any commuting family of nonexpansive maps (𝑇𝑖)π‘–βˆˆπΌ, with π‘‡π‘–βˆΆ(𝑋,𝑑)β†’(𝑋,𝑑), has a common fixed point. Moreover, the common fixed point set ξ™π‘–βˆˆπΌξ€·π‘‡Fix𝑖,(4.2) is π‘ž-hyperconvex.

Proof. We observe that (𝑋,𝑑𝑠) is a bounded hyperconvex metric space by [1, Proposition 2], and for each π‘–βˆˆπΌ, the map π‘‡π‘–βˆΆ(𝑋,𝑑𝑠)β†’(𝑋,𝑑𝑠) is nonexpansive, as we noted before (see proof of Theorem 3.3). By Theorem 3.3, each 𝑇𝑖 has a fixed point. Hence, there is π‘₯βˆˆπ‘‹ such that 𝑇𝑖(π‘₯)=π‘₯. We now show that, given any π‘—βˆˆπΌ, we have that 𝑇𝑗(Fix(𝑇𝑖))βŠ†Fix(𝑇𝑖) Indeed, if for some π‘₯βˆˆπ‘‹, we have π‘₯=𝑇𝑖(π‘₯), then 𝑇𝑗(π‘₯)=𝑇𝑗(𝑇𝑖(π‘₯))=𝑇𝑖(𝑇𝑗(π‘₯)). So 𝑇𝑗(π‘₯)∈Fix(𝑇𝑖).
By Theorem 3.3, we conclude that π‘‡π‘—βˆΆFix(𝑇𝑖)β†’Fix(𝑇𝑖) has a fixed point π‘¦βˆˆFix(𝑇𝑖), which then is a fixed point of 𝑇𝑖 and 𝑇𝑗. Therefore, the set of common fixed points of 𝑇𝑖 and 𝑇𝑗 is π‘ž-hyperconvex by Theorem 3.3. Hence, by induction for each finite family (𝑇𝑖)π‘–βˆˆπΉ of nonexpansive self-maps on 𝑋 the set of common fixed points is nonempty and π‘ž-hyperconvex.
Since β‹‚π‘–βˆˆπΉFix(𝑇𝑖) is nonempty subset and π‘ž-hyperconvex whenever 𝐹 is a finite subset of 𝐼, by Lemma 4.3 we conclude that β‹‚π‘–βˆˆπΌFix(𝑇𝑖) is nonempty and π‘ž-hyperconvex.

5. Approximate Fixed Points

In this section, we investigate the approximation of fixed points by generalizing some results from [13] (compare [18]). We first define an πœ–1,πœ–2-parallel set of a subset in a quasipseudometric space similarly to [13, page 89].

Definition 5.1. Let (𝑋,𝑑) be a quasipseudometric space. Given a subset 𝐴 of 𝑋, we define for πœ–1,πœ–2β‰₯0 the πœ–1,πœ–2-parallel set of 𝐴 as π‘πœ–1,πœ–2(𝐴)=π‘Žβˆˆπ΄ξ€·πΆπ‘‘ξ€·π‘Ž,πœ–2ξ€Έβˆ©πΆπ‘‘βˆ’1ξ€·π‘Ž,πœ–1.ξ€Έξ€Έ(5.1)
(Note that for each πœ–>0 in particular π‘πœ–,πœ–β‹ƒ(𝐴)=π‘Žβˆˆπ΄πΆπ‘‘π‘ (π‘Ž,πœ–).)
Thus, π‘₯βˆˆπ‘πœ–1,πœ–2(𝐴) if and only if there exists π‘Žβˆˆπ΄ such that 𝑑(π‘Ž,π‘₯)β‰€πœ–2 and π‘‘βˆ’1(π‘Ž,π‘₯)β‰€πœ–1.
We next give a characterization of π‘πœ–1,πœ–2(𝐴) if A is a π‘ž-admissible set in a π‘ž-hyperconvex quasipseudometric space (compare [13, Lemma 4.2]).

Lemma 5.2. Let (𝑋,𝑑) be a π‘ž-hyperconvex quasipseudometric space and let 𝐴 be a π‘ž-admissible subset of 𝑋, say β‹‚βˆ…β‰ π΄=π‘–βˆˆπΌ(𝐢𝑑(π‘₯𝑖,π‘Ÿπ‘–)βˆ©πΆπ‘‘βˆ’1(π‘₯𝑖,𝑠𝑖)) with π‘₯π‘–βˆˆπ‘‹ and π‘Ÿπ‘–,𝑠𝑖 nonnegative reals whenever π‘–βˆˆπΌβ‰ βˆ…. Then, for each πœ–1,πœ–2β‰₯0, π‘πœ–1,πœ–2(𝐴)=π‘–βˆˆπΌξ€·πΆπ‘‘ξ€·π‘₯𝑖,π‘Ÿπ‘–+πœ–2ξ€Έβˆ©πΆπ‘‘βˆ’1ξ€·π‘₯𝑖,𝑠𝑖+πœ–1.ξ€Έξ€Έ(5.2)

Proof. Suppose that π‘¦βˆˆπ‘πœ–1,πœ–2(𝐴). Then, 𝑑(π‘Ž,𝑦)β‰€πœ–2 and 𝑑(𝑦,π‘Ž)β‰€πœ–1 for some π‘Žβˆˆπ΄. But for each π‘–βˆˆπΌ, 𝑑π‘₯𝑖π‘₯,𝑦≀𝑑𝑖,π‘Ž+𝑑(π‘Ž,𝑦)β‰€π‘Ÿπ‘–+πœ–2,𝑑𝑦,π‘₯𝑖≀𝑑(𝑦,π‘Ž)+π‘‘π‘Ž,π‘₯π‘–ξ€Έβ‰€πœ–1+𝑠𝑖.(5.3) Then, for each π‘–βˆˆπΌ, we have π‘¦βˆˆπΆπ‘‘(π‘₯𝑖,π‘Ÿπ‘–+πœ–2) and π‘¦βˆˆπΆπ‘‘βˆ’1(π‘₯𝑖,𝑠𝑖+πœ–1). This proves that π‘πœ–1,πœ–2β‹‚(𝐴)βŠ†π‘–βˆˆπΌ(𝐢𝑑(π‘₯𝑖,π‘Ÿπ‘–+πœ–2)βˆ©πΆπ‘‘βˆ’1(π‘₯𝑖,𝑠𝑖+πœ–1)).
Now suppose that β‹‚π‘¦βˆˆπ‘–βˆˆπΌ(𝐢𝑑(π‘₯𝑖,π‘Ÿπ‘–+πœ–2)βˆ©πΆπ‘‘βˆ’1(π‘₯𝑖,𝑠𝑖+πœ–1)) and let π‘–βˆˆπΌ.
Hence, 𝑑π‘₯𝑖,π‘¦β‰€π‘Ÿπ‘–+πœ–2,𝑑𝑦,π‘₯π‘–ξ€Έβ‰€πœ–1+𝑠𝑖.(5.4) By definition of 𝐴 and the triangle inequality, for any π‘Žβˆˆπ΄ and any 𝑖,π‘—βˆˆπΌ we must have that 𝑑π‘₯𝑖,π‘₯𝑗π‘₯≀𝑑𝑖,π‘Ž+π‘‘π‘Ž,π‘₯π‘—ξ€Έβ‰€π‘Ÿπ‘–+𝑠𝑗.(5.5) Hence, [(𝐢𝑑(π‘₯𝑖,π‘Ÿπ‘–))π‘–βˆˆπΌ,𝐢𝑑(𝑦,πœ–1);(πΆπ‘‘βˆ’1(π‘₯𝑖,𝑠𝑖))π‘–βˆˆπΌ,πΆπ‘‘βˆ’1(𝑦,πœ–2)] satisfies the hypothesis in the definition of π‘ž-hyperconvexity of (𝑋,𝑑).
So, by π‘ž-hyperconvexity of 𝑋, ξƒ©ξ™βˆ…β‰ π‘–βˆˆπΌπΆπ‘‘ξ€·π‘₯𝑖,π‘Ÿπ‘–ξ€Έξƒͺβˆ©πΆπ‘‘ξ€·π‘¦,πœ–1ξ€Έβˆ©ξƒ©ξ™π‘–βˆˆπΌπΆπ‘‘βˆ’1ξ€·π‘₯𝑖,𝑠𝑖ξƒͺβˆ©πΆπ‘‘βˆ’1𝑦,πœ–2ξ€Έ=ξ™π‘–βˆˆπΌξ€·πΆπ‘‘ξ€·π‘₯𝑖,π‘Ÿπ‘–ξ€Έβˆ©πΆπ‘‘βˆ’1ξ€·π‘₯𝑖,π‘ π‘–βˆ©ξ€·πΆξ€Έξ€Έπ‘‘ξ€·π‘¦,πœ–1ξ€Έβˆ©πΆπ‘‘βˆ’1𝑦,πœ–2𝐢=π΄βˆ©π‘‘ξ€·π‘¦,πœ–1ξ€Έβˆ©πΆπ‘‘βˆ’1𝑦,πœ–2.ξ€Έξ€Έ(5.6)
Therefore, there is π‘Žβˆˆπ΄ such that 𝑑(𝑦,π‘Ž)β‰€πœ–1 and 𝑑(π‘Ž,𝑦)β‰€πœ–2. Hence, π‘¦βˆˆπ‘πœ–1,πœ–2(𝐴) and the proof is complete.

The following lemma will be needed in our discussion below of approximate fixed point sets.

Lemma 5.3 (compare [13, Lemma 4.3]). Suppose that (𝑋,𝑑) is a π‘ž-hyperconvex 𝑇0-quasimetric space and let 𝐴 be a π‘ž-admissible subset of  𝑋. Then, for each πœ–1,πœ–2β‰₯0 there is a nonexpansive retraction 𝑅 of π‘πœ–1,πœ–2(𝐴) onto 𝐴 which has the property that 𝑑(π‘₯,𝑅(π‘₯))β‰€πœ–1 and 𝑑(𝑅(π‘₯),π‘₯)β‰€πœ–2 whenever π‘₯βˆˆπ‘πœ–1,πœ–2(𝐴).

Proof. Assume β‹‚βˆ…β‰ π΄=π‘–βˆˆπΌ(𝐢𝑑(π‘₯𝑖,π‘Ÿπ‘–)βˆ©πΆπ‘‘βˆ’1(π‘₯𝑖,𝑠𝑖)) with πΌβ‰ βˆ…. By Lemma 5.2, we know that π‘πœ–1,πœ–2(𝐴) is π‘ž-admissible in (𝑋,𝑑) and so π‘πœ–1,πœ–2(𝐴) is itself π‘ž-hyperconvex by Proposition 3.1. Consider the family β„±={(𝐷,𝑅𝐷)βˆΆπ΄βŠ†π·βŠ†π‘πœ–1,πœ–2(𝐴) and π‘…π·βˆΆπ·β†’π΄ is a nonexpansive retraction such that 𝑑(π‘₯,𝑅(π‘₯))β‰€πœ–1 and 𝑑(𝑅(π‘₯),π‘₯)β‰€πœ–2 whenever π‘₯∈𝐷}.
Let Id denote the identity map on 𝐴. Note that (𝐴,Id)βˆˆβ„±. So β„±β‰ βˆ…. If one partially orders β„± in the usual way ((𝐷,𝑅𝐷)β‰Ό(𝐻,𝑅𝐻) if and only if π·βŠ†π» and 𝑅𝐻 is an extension of 𝑅𝐷), then each chain in (β„±,β‰Ό) is bounded above. So by Zorn's lemma β„± has a maximal element which we denote by (𝐷,𝑅𝐷). We need to show that 𝐷=π‘πœ–1,πœ–2(𝐴). Suppose that there exists π‘₯βˆˆπ‘πœ–1,πœ–2(𝐴) such that π‘₯βˆ‰π·, and consider the set 𝐢=π‘€βˆˆπ·ξ€·πΆπ‘‘ξ€·π‘…π·ξ€Έ(𝑀),𝑑(𝑀,π‘₯)βˆ©πΆπ‘‘βˆ’1ξ€·π‘…π·ξƒ­βˆ©ξƒ¬ξ™(𝑀),𝑑(π‘₯,𝑀)ξ€Έξ€Έπ‘–βˆˆπΌξ€·πΆπ‘‘ξ€·π‘₯𝑖,π‘Ÿπ‘–ξ€Έβˆ©πΆπ‘‘βˆ’1ξ€·π‘₯𝑖,π‘ π‘–ξƒ­βˆ©ξ€ΊπΆξ€Έξ€Έπ‘‘ξ€·π‘₯,πœ–1ξ€Έβˆ©πΆπ‘‘βˆ’1ξ€·π‘₯,πœ–2.ξ€Έξ€»(5.7) First, we want to show that πΆβ‰ βˆ…, and in order to do this by [1, Proposition 1], we need only to show that the family 𝐢𝑑𝑅𝐷(𝑀),𝑑(𝑀,π‘₯)ξ€Έξ€Έπ‘€βˆˆπ·,𝐢𝑑π‘₯𝑖,π‘Ÿπ‘–ξ€Έξ€Έπ‘–βˆˆπΌ,𝐢𝑑π‘₯,πœ–1ξ€Έ;ξ€·πΆπ‘‘βˆ’1𝑅𝐷(𝑀),𝑑(π‘₯,𝑀)ξ€Έξ€Έπ‘€βˆˆπ·,ξ€·πΆπ‘‘βˆ’1ξ€·π‘₯𝑖,π‘ π‘–ξ€Έξ€Έπ‘–βˆˆπΌ,πΆπ‘‘βˆ’1ξ€·π‘₯,πœ–2ξ€Έξ€»(5.8) of double balls has the mixed binary intersection property.
First note that if 𝑀1,𝑀2∈𝐷, then 𝑑𝑅𝐷𝑀1ξ€Έ,𝑅𝐷𝑀2𝑀≀𝑑1,𝑀2𝑀≀𝑑1ξ€Έξ€·,π‘₯+𝑑π‘₯,𝑀2ξ€Έ.(5.9) Therefore, 𝐢𝑑(𝑅𝐷(𝑀1),𝑑(𝑀1,π‘₯)) and πΆπ‘‘βˆ’1(𝑅𝐷(𝑀2),𝑑(π‘₯,𝑀2)) intersect by metric convexity of (𝑋,𝑑).
Furthermore, by the definition of 𝐴, for each 𝑖,π‘—βˆˆπΌ, we see that 𝐢𝑑(π‘₯𝑖,π‘Ÿπ‘–) and πΆπ‘‘βˆ’1(π‘₯𝑗,𝑠𝑗) intersect.
Also for each π‘€βˆˆπ·, 𝑅𝐷⋂(𝑀)∈𝐴=π‘–βˆˆπΌ(𝐢𝑑(π‘₯𝑖,π‘Ÿπ‘–)βˆ©πΆπ‘‘βˆ’1(π‘₯𝑖,𝑠𝑖)). Hence, for any π‘€βˆˆπ· and π‘–βˆˆπΌ,𝐢𝑑(𝑅𝐷(𝑀),𝑑(𝑀,π‘₯)) and πΆπ‘‘βˆ’1(π‘₯𝑖,𝑠𝑖) intersect, as well as for any π‘€βˆˆπ· and π‘–βˆˆπΌ,πΆπ‘‘βˆ’1(𝑅𝐷(𝑀),𝑑(π‘₯,𝑀)) and 𝐢𝑑(π‘₯𝑖,π‘Ÿπ‘–) intersect.
Since π‘₯βˆˆπ‘πœ–1,πœ–2(𝐴)=π‘–βˆˆπΌξ€·πΆπ‘‘ξ€·π‘₯𝑖,π‘Ÿπ‘–+πœ–2ξ€Έβˆ©πΆπ‘‘βˆ’1ξ€·π‘₯𝑖,𝑠𝑖+πœ–1,ξ€Έξ€Έ(5.10) by Lemma 5.2 there is π‘Žβˆˆπ΄ such that π‘₯βˆˆπΆπ‘‘(π‘Ž,πœ–2)βˆ©πΆπ‘‘βˆ’1(π‘Ž,πœ–1) and, therefore, (𝐢𝑑(π‘₯,πœ–1)βˆ©πΆπ‘‘βˆ’1(π‘₯,πœ–2))∩(𝐢𝑑(π‘₯𝑖,π‘Ÿπ‘–)βˆ©πΆπ‘‘βˆ’1(π‘₯𝑖,𝑠𝑖))β‰ βˆ… whenever π‘–βˆˆπΌ.
Finally, if π‘€βˆˆπ·, then by assumption on 𝑅𝐷, 𝑑𝑅𝐷𝑅(𝑀),π‘₯≀𝑑𝐷(𝑀),𝑀+𝑑(𝑀,π‘₯)β‰€πœ–2𝑑+𝑑(𝑀,π‘₯),π‘₯,𝑅𝐷(𝑀)≀𝑑(π‘₯,𝑀)+𝑑𝑀,𝑅𝐷(𝑀)≀𝑑(π‘₯,𝑀)+πœ–1.(5.11) Thus, by metric convexity of (𝑋,𝑑), we have that 𝐢𝑑(𝑅𝐷(𝑀),𝑑(𝑀,π‘₯)) and πΆπ‘‘βˆ’1(π‘₯,πœ–2) intersect as well as πΆπ‘‘βˆ’1(𝑅𝐷(𝑀),𝑑(π‘₯,𝑀)) and 𝐢𝑑(π‘₯,πœ–1) intersect.
Of course, 𝐢𝑑(π‘₯,πœ–1) and πΆπ‘‘βˆ’1(π‘₯,πœ–2) intersect.
We have shown that the family 𝐢𝑑𝑅𝐷(𝑀),𝑑(𝑀,π‘₯)π‘€βˆˆπ·,𝐢𝑑π‘₯𝑖,π‘Ÿπ‘–ξ€Έξ€Έπ‘–βˆˆπΌ,𝐢𝑑π‘₯,πœ–1ξ€Έ;πΆπ‘‘βˆ’1𝑅𝐷(𝑀),𝑑(π‘₯,𝑀)π‘€βˆˆπ·,ξ€·πΆπ‘‘βˆ’1ξ€·π‘₯𝑖,π‘ π‘–ξ€Έξ€Έπ‘–βˆˆπΌ,πΆπ‘‘βˆ’1ξ€·π‘₯,πœ–2ξ€Έξ€»(5.12) of double balls has the mixed binary intersection property.
Hence, βˆ…β‰ πΆβŠ†π΄. Now, let π‘’βˆˆπΆ and define π‘…ξ…žβˆΆπ·βˆͺ{π‘₯}→𝐴 by setting π‘…ξ…ž(𝑀)=𝑅𝐷(𝑀) if π‘€βˆˆπ· and π‘…ξ…ž(π‘₯)=𝑒. Then, for each π‘€βˆˆπ·, we have π‘‘ξ€·π‘…ξ…ž(π‘₯),π‘…ξ…žξ€Έξ€·(𝑀)=𝑑𝑒,𝑅𝐷𝑑𝑅(𝑀)≀𝑑(π‘₯,𝑀),ξ…ž(𝑀),π‘…ξ…ž(𝑅π‘₯)=𝑑𝐷(𝑀),𝑒≀𝑑(𝑀,π‘₯),(5.13) so that π‘…ξ…ž is nonexpansive. Also, 𝑑(π‘…ξ…ž(π‘₯),π‘₯)=𝑑(𝑒,π‘₯)β‰€πœ–2 and 𝑑(π‘₯,π‘…ξ…ž(π‘₯))=𝑑(π‘₯,𝑒)β‰€πœ–1. Therefore, we conclude that the pair (𝐷βˆͺ{π‘₯},π‘…ξ…ž) contradicts the maximality of (𝐷,𝑅𝐷) in (β„±,β‰Ό). Consequently, 𝐷=π‘πœ–1,πœ–2(𝐴) and we are done.

Definition 5.4 (compare [19] and [20]). Let (𝑋,𝑑) be a 𝑇0-quasimetric space. We say that a map π‘‡βˆΆ(𝑋,𝑑)β†’(𝑋,𝑑) has approximate fixed points if infπ‘₯βˆˆπ‘‹π‘‘π‘ (π‘₯,𝑇(π‘₯))=0.

Definition 5.5. Let (𝑋,𝑑) be a 𝑇0-quasimetric space. For a map π‘‡βˆΆ(𝑋,𝑑)β†’(𝑋,𝑑) and for any πœ–1,πœ–2β‰₯0, we use πΉπœ–1,πœ–2(𝑇) to denote the set of πœ–1,πœ–2-approximate fixed points of 𝑇; that is, πΉπœ–1,πœ–2(𝑇)={π‘₯βˆˆπ‘‹βˆΆπ‘‘(π‘₯,𝑇(π‘₯))β‰€πœ–2 and 𝑑(𝑇(π‘₯),π‘₯)β‰€πœ–1}.

Theorem 5.6 (compare [13, Theorem 4.11]). Suppose that (𝑋,𝑑) is a π‘ž-hyperconvex 𝑇0-quasimetric space and that the map π‘‡βˆΆ(𝑋,𝑑)β†’(𝑋,𝑑) is nonexpansive. Furthermore suppose that for some πœ–1,πœ–2β‰₯0 one has that πΉπœ–1,πœ–2(𝑇) is nonempty. Then, the set πΉπœ–1,πœ–2(𝑇) is π‘ž-hyperconvex.

Proof. For each 𝑖 in some nonempty index set 𝐼, let π‘₯π‘–βˆˆπΉπœ–1,πœ–2(𝑇), and let π‘Ÿπ‘–β‰₯0 and 𝑠𝑖β‰₯0 satisfy 𝑑π‘₯𝑖,π‘₯π‘—ξ€Έβ‰€π‘Ÿπ‘–+𝑠𝑗.(5.14) We need to show that ξƒ¬ξ™π‘–βˆˆπΌξ€·πΆπ‘‘ξ€·π‘₯𝑖,π‘Ÿπ‘–ξ€Έβˆ©πΆπ‘‘βˆ’1ξ€·π‘₯𝑖,π‘ π‘–ξƒ­ξ€Έξ€Έβˆ©πΉπœ–1,πœ–2(𝑇)β‰ βˆ….(5.15) We know that β‹‚βˆ…β‰ π½=π‘–βˆˆπΌ(𝐢𝑑(π‘₯𝑖,π‘Ÿπ‘–)βˆ©πΆπ‘‘βˆ’1(π‘₯𝑖,𝑠𝑖)) is π‘ž-hyperconvex according to Proposition 3.1, since (𝑋,𝑑) is π‘ž-hyperconvex. Furthermore, 𝐽 is obviously bounded in (𝑋,𝑑).
Also, if π‘₯∈𝐽, then for each π‘–βˆˆπΌ, 𝑑π‘₯𝑖π‘₯,𝑇(π‘₯)≀𝑑𝑖π‘₯,𝑇𝑖𝑇π‘₯ξ€Έξ€Έ+𝑑𝑖,𝑇(π‘₯)β‰€πœ–2ξ€·π‘₯+𝑑𝑖,π‘₯β‰€πœ–2+π‘Ÿπ‘–,𝑑𝑇(π‘₯),π‘₯𝑖π‘₯≀𝑑𝑇(π‘₯),𝑇𝑖𝑇π‘₯ξ€Έξ€Έ+𝑑𝑖,π‘₯𝑖≀𝑑π‘₯,π‘₯𝑖+πœ–1≀𝑠𝑖+πœ–1.(5.16)
This proves that 𝑇(π‘₯)βˆˆπ‘πœ–1,πœ–2(𝐽) by Lemma 5.2. Now, by Lemma 5.3, there is a nonexpansive retraction 𝑅 of π‘πœ–1,πœ–2(𝐽) onto 𝐽 for which 𝑑(𝑅(π‘₯),π‘₯)β‰€πœ–2 and 𝑑(π‘₯,𝑅(π‘₯))β‰€πœ–1 whenever π‘₯βˆˆπ‘πœ–1,πœ–2(𝐽). Also since π‘…βˆ˜π‘‡ is a nonexpansive map of 𝐽 into 𝐽, it must have a fixed point by Theorem 3.3.
Suppose that (π‘…βˆ˜π‘‡)(π‘₯0)=π‘₯0 for some π‘₯0∈𝐽. Then, 𝑑π‘₯0ξ€·π‘₯,𝑇0ξ€·ξ€·π‘₯ξ€Έξ€Έ=𝑑(π‘…βˆ˜π‘‡)0ξ€Έξ€·π‘₯,𝑇0ξ€Έξ€Έβ‰€πœ–2,𝑑𝑇π‘₯0ξ€Έ,π‘₯0𝑇π‘₯=𝑑0ξ€Έξ€·π‘₯,(π‘…βˆ˜π‘‡)0ξ€Έξ€Έβ‰€πœ–1.(5.17) Thus, the proof is complete, since π‘₯0βˆˆπ½βˆ©πΉπœ–1,πœ–2(𝑇).

6. External π‘ž-Hyperconvexity

We next define an externally π‘ž-hyperconvex subset of a quasipseudometric space (𝑋,𝑑) in analogy to [17, Definition 3.5]. Note that this definition strengthens the concept of a π‘ž-hyperconvex subset of (𝑋,𝑑) (compare also [21, Definition 3]).

Definition 6.1. Let (𝑋,𝑑) be a quasipseudometric space. A subspace 𝐸 of (𝑋,𝑑) is said to be externally π‘ž-hyperconvex (relative to 𝑋) if given any family (π‘₯𝑖)π‘–βˆˆπΌ of points in 𝑋 and families of nonnegative real numbers (π‘Ÿπ‘–)π‘–βˆˆπΌ and (𝑠𝑖)π‘–βˆˆπΌ the following condition holds: if 𝑑(π‘₯𝑖,π‘₯𝑗)β‰€π‘Ÿπ‘–+𝑠𝑗 whenever 𝑖,π‘—βˆˆπΌ, dist(π‘₯𝑖,𝐸)β‰€π‘Ÿπ‘– and dist(𝐸,π‘₯𝑖)≀𝑠𝑖 whenever π‘–βˆˆπΌ, then β‹‚π‘–βˆˆπΌ(𝐢𝑑(π‘₯𝑖,π‘Ÿπ‘–)βˆ©πΆπ‘‘βˆ’1(π‘₯𝑖,𝑠𝑖))βˆ©πΈβ‰ βˆ….
In the following, β„°π‘ž(𝑋) will denote the set of nonempty externally π‘ž-hyperconvex subsets of (𝑋,𝑑).

Example 6.2 (compare [21, Theorem 7]). Let 𝐸 be a nonempty externally π‘ž-hyperconvex subset in a quasipseudometric space (𝑋,𝑑) and let π‘₯ be any point of 𝑋. Set dist(π‘₯,𝐸)=π‘Ÿ and dist(𝐸,π‘₯)=𝑠. Then, by applying external π‘ž-hyperconvexity of 𝐸 to the double ball (𝐢𝑑(π‘₯,π‘Ÿ);πΆπ‘‘βˆ’1(π‘₯,𝑠)), we conclude that there is π‘βˆˆπΆπ‘‘(π‘₯,π‘Ÿ)βˆ©πΆπ‘‘βˆ’1(π‘₯,𝑠)∩𝐸. Thus, 𝑑(π‘₯,𝑝)=dist(π‘₯,𝐸) and 𝑑(𝑝,π‘₯)=dist(𝐸,π‘₯).

Lemma 6.3 (compare [17, Lemma 3.8]). Let (𝑋,𝑑) be a π‘ž-hyperconvex space and let π‘₯βˆˆπ‘‹. Furthermore, let β‹‚βˆ…β‰ π΄=π‘–βˆˆπΌ(𝐢𝑑(π‘₯𝑖,π‘Ÿπ‘–)βˆ©πΆπ‘‘βˆ’1(π‘₯𝑖,𝑠𝑖)) where (π‘₯𝑖)π‘–βˆˆπΌ is a nonempty family of points in 𝑋 and (π‘Ÿπ‘–)π‘–βˆˆπΌ and (𝑠𝑖)π‘–βˆˆπΌ are families of nonnegative reals. Then, there is π‘βˆˆπ΄ such that dist(π‘₯,𝐴)=𝑑(π‘₯,𝑝) and dist(𝐴,π‘₯)=𝑑(𝑝,π‘₯).

Proof. Evidently, 𝐢𝑑π‘₯𝑖,π‘Ÿπ‘–ξ€Έξ€Έπ‘–βˆˆπΌ,𝐢𝑑(π‘₯,dist(π‘₯,𝐴)+πœ–)πœ–>0;(πΆπ‘‘βˆ’1(π‘₯𝑖,𝑠𝑖))π‘–βˆˆπΌ,(πΆπ‘‘βˆ’1(π‘₯,dist(𝐴,π‘₯)+πœ–))πœ–>0ξ€»(6.1) satisfies the mixed binary intersection property. Thus, there is π‘βˆˆπ΄βˆ©πΆπ‘‘(π‘₯,dist(π‘₯,𝐴))βˆ©πΆπ‘‘βˆ’1(π‘₯,dist(𝐴,π‘₯))(6.2) by π‘ž-hyperconvexity of (𝑋,𝑑). Obviously, 𝑝 then satisfies the stated condition.

The following lemma, which makes use of Lemma 6.3, will be useful in the proof of Theorem 6.5. Considering the case that 𝐸=𝑋, we see that Lemma 6.4 improves on Proposition 3.1.

Lemma 6.4 (compare [18, Lemma 2]). Let (𝑋,𝑑) be a π‘ž-hyperconvex quasipseudometric space. Suppose that πΈβŠ†π‘‹ is externally π‘ž-hyperconvex relative to 𝑋 and suppose that 𝐴 is a π‘ž-admissible subset of (𝑋,𝑑) such that πΈβˆ©π΄β‰ βˆ…. Then 𝐸∩𝐴 is externally π‘ž-hyperconvex relative to 𝑋.

Proof. Assume that a given nonempty family (π‘₯𝛼)π›Όβˆˆπ‘† of points in 𝑋 and families of nonnegative real numbers (π‘Ÿπ›Ό)π›Όβˆˆπ‘† and (𝑠𝛼)π›Όβˆˆπ‘† satisfy 𝑑(π‘₯𝛼,π‘₯𝛽)β‰€π‘Ÿπ›Ό+𝑠𝛽, dist(π‘₯𝛼,𝐴∩𝐸)β‰€π‘Ÿπ›Ό, and dist(𝐴∩𝐸,π‘₯𝛼)≀𝑠𝛼 whenever 𝛼,π›½βˆˆπ‘†.
Since 𝐴 is π‘ž-admissible, β‹‚βˆ…β‰ π΄=π‘–βˆˆπΌ(𝐢𝑑(π‘₯𝑖,π‘Ÿπ‘–)βˆ©πΆπ‘‘βˆ’1(π‘₯𝑖,𝑠𝑖)) with π‘₯π‘–βˆˆπ‘‹ and π‘Ÿπ‘–,𝑠𝑖β‰₯0 whenever π‘–βˆˆπΌ. Because dist(π‘₯𝛼,𝐴)≀dist(π‘₯𝛼,𝐴∩𝐸)β‰€π‘Ÿπ›Ό and dist(𝐴,π‘₯𝛼)≀dist(𝐴∩𝐸,π‘₯𝛼)≀𝑠𝛼 whenever π›Όβˆˆπ‘†, it follows that for eachβ€‰β€‰π›Όβˆˆπ‘†, π‘–βˆˆπΌ and for π‘βˆˆπ΄ chosen according to Lemma 6.3 we have 𝑑π‘₯𝛼,π‘₯𝑖π‘₯≀𝑑𝛼,𝑝+𝑑𝑝,π‘₯π‘–ξ€Έβ‰€π‘Ÿπ›Ό+𝑠𝑖,𝑑π‘₯𝑖,π‘₯𝛼π‘₯≀𝑑𝑖,𝑝+𝑑𝑝,π‘₯π›Όξ€Έβ‰€π‘Ÿπ‘–+𝑠𝛼.(6.3) Also, since for each π‘–βˆˆπΌ, π΄βŠ†πΆπ‘‘(π‘₯𝑖,π‘Ÿπ‘–)βˆ©πΆπ‘‘βˆ’1(π‘₯𝑖,𝑠𝑖), and since π΄βˆ©πΈβ‰ βˆ…, it follows that ξ€·π‘₯dist𝑖,πΈβ‰€π‘Ÿπ‘–,ξ€·dist𝐸,π‘₯𝑖≀𝑠𝑖,(6.4) and that 𝑑(π‘₯𝑖,π‘₯𝑗)β‰€π‘Ÿπ‘–+𝑠𝑗 whenever 𝑖,π‘—βˆˆπΌ. Trivially, we have dist(π‘₯𝛼,𝐸)β‰€π‘Ÿπ›Ό and dist(𝐸,π‘₯𝛼)≀𝑠𝛼 whenever π›Όβˆˆπ‘†.
Therefore, by external π‘ž-hyperconvexity of 𝐸, we conclude that ξƒ¬ξ™π‘–βˆˆπΌξ€·πΆπ‘‘ξ€·π‘₯𝑖,π‘Ÿπ‘–ξ€Έβˆ©πΆπ‘‘βˆ’1ξ€·π‘₯𝑖,π‘ π‘–ξƒ­βˆ©ξƒ¬ξ™ξ€Έξ€Έπ›Όβˆˆπ‘†ξ€·πΆπ‘‘ξ€·π‘₯𝛼,π‘Ÿπ›Όξ€Έβˆ©πΆπ‘‘βˆ’1ξ€·π‘₯𝛼,𝑠𝛼=ξ™ξ€Έξ€Έβˆ©πΈπ›Όβˆˆπ‘†ξ€·πΆπ‘‘ξ€·π‘₯𝛼,π‘Ÿπ›Όξ€Έβˆ©πΆπ‘‘βˆ’1ξ€·π‘₯𝛼,π‘ π›Όξ€Έξ€Έβˆ©(𝐸∩𝐴)β‰ βˆ….(6.5) Thus, the proof is complete.

We next show that the intersection of a descending family of externally π‘ž-hyperconvex nonempty subspaces of a bounded π‘ž-hyperconvex 𝑇0-quasimetric space behaves as expected.

Theorem 6.5 (compare [18, Theorem 4]). Let (𝑋,𝑑) be a bounded π‘ž-hyperconvex 𝑇0-quasimetric space. Moreover, let (𝑋𝑖)π‘–βˆˆπΌ be a descending family of nonempty externally π‘ž-hyperconvex subsets of 𝑋, where 𝐼 is assumed to be totally ordered such that 𝑖1,𝑖2∈𝐼 and 𝑖1≀𝑖2 if and only if 𝑋𝑖2βŠ†π‘‹π‘–1. Thenβ‹‚π‘–βˆˆπΌπ‘‹π‘– is nonempty and externally π‘ž-hyperconvex relative to 𝑋.

Proof. Theorem 4.1 implies that ⋂𝐷=π‘–βˆˆπΌπ‘‹π‘–β‰ βˆ…. In order to show that 𝐷 is externally π‘ž-hyperconvex, let a nonempty family (π‘₯𝛼)π›Όβˆˆπ‘† of points in 𝑋 and families of nonnegative real numbers (π‘Ÿπ›Ό)π›Όβˆˆπ‘† and (𝑠𝛼)π›Όβˆˆπ‘† be given such that 𝑑(π‘₯𝛼,π‘₯𝛽)β‰€π‘Ÿπ›Ό+𝑠𝛽, and dist(π‘₯𝛼,𝐷)β‰€π‘Ÿπ›Ό and dist(𝐷,π‘₯𝛼)≀𝑠𝛼 whenever 𝛼,π›½βˆˆπ‘†.
Since 𝑋 is π‘ž-hyperconvex, we know that β‹‚π΄βˆΆ=π›Όβˆˆπ‘†(𝐢𝑑(π‘₯𝛼,π‘Ÿπ›Ό)βˆ©πΆπ‘‘βˆ’1(π‘₯𝛼,𝑠𝛼))β‰ βˆ…. Also, since for each π›Όβˆˆπ‘†, dist(π‘₯𝛼,𝐷)β‰€π‘Ÿπ›Ό and dist(𝐷,π‘₯𝛼)≀𝑠𝛼, we have dist(π‘₯𝛼,𝑋𝑖)β‰€π‘Ÿπ›Ό and dist(𝑋𝑖,π‘₯𝛼)≀𝑠𝛼 whenever π‘–βˆˆπΌ, so that, by external π‘ž-hyperconvexity of 𝑋𝑖, we conclude that π΄βˆ©π‘‹π‘–β‰ βˆ… whenever π‘–βˆˆπΌ.
By Lemma 6.4,  (π΄βˆ©π‘‹π‘–)π‘–βˆˆπΌ is a descending chain of nonempty (externally) π‘ž-hyperconvex subsets of (𝑋,𝑑), so that again by Theorem 4.1β€‰β€‰β‹‚π‘–βˆˆπΌ(π΄βˆ©π‘‹π‘–)=π΄βˆ©π·β‰ βˆ….

Let us note that the result stated in our abstract is a consequence of our next theorem.

Theorem 6.6 (compare [18, Theorem 1]). Let (𝐻,𝑑) be a π‘ž-hyperconvex 𝑇0-quasimetric space, let 𝑋 be any set, and let a map π‘‡βˆ—βˆΆπ‘‹β†’β„°π‘ž(𝐻) be given. Then, there exists a map π‘‡βˆΆπ‘‹β†’π» for which 𝑇(π‘₯)βˆˆπ‘‡βˆ—(π‘₯) whenever π‘₯βˆˆπ‘‹ and for which 𝑑(𝑇(π‘₯),𝑇(𝑦))≀𝑑𝐻(π‘‡βˆ—(π‘₯),π‘‡βˆ—(𝑦)) whenever π‘₯,π‘¦βˆˆπ‘‹.

Proof. Let β„± denote the collection of all pairs (𝐷,𝑇), where π·βŠ†π‘‹,π‘‡βˆΆπ·β†’π»,𝑇(𝑑)βˆˆπ‘‡βˆ—(𝑑) whenever π‘‘βˆˆπ·, and 𝑑(𝑇(π‘₯),𝑇(𝑦))≀𝑑𝐻(π‘‡βˆ—(π‘₯),π‘‡βˆ—(𝑦)) whenever π‘₯,π‘¦βˆˆπ·. Note that β„±β‰ βˆ…, since ({π‘₯0},𝑇)βˆˆβ„± for any choice of π‘₯0βˆˆπ‘‹ and 𝑇(π‘₯0)βˆˆπ‘‡βˆ—(π‘₯0). Define a partial order relation on β„± by setting (𝐷1,𝑇1)β‰Ό(𝐷2,𝑇2) if and only if 𝐷1βŠ†π·2, and, 𝑇2|𝐷1=𝑇1.
Let ((𝐷𝛼,𝑇𝛼))π›Όβˆˆπ‘† be an increasing chain in (β„±,β‰Ό). Then it follows that (β‹ƒπ›Όβˆˆπ‘†π·π›Ό,𝑇)βˆˆβ„± where 𝑇|𝐷𝛼=𝑇𝛼. By Zorn's lemma, (β„±,β‰Ό) has a maximal element, say (𝐷,𝑇). Assume that 𝐷≠𝑋 and select π‘₯0βˆˆπ‘‹β§΅π·. Set 𝐷=𝐷βˆͺ{π‘₯0} and consider the set 𝐽=π‘₯βˆˆπ·ξ€ΊπΆπ‘‘ξ€·π‘‡(π‘₯),π‘‘π»ξ€·π‘‡βˆ—(π‘₯),π‘‡βˆ—ξ€·π‘₯0ξ€Έξ€Έξ€Έβˆ©πΆπ‘‘βˆ’1𝑇(π‘₯),π‘‘π»ξ€·π‘‡βˆ—ξ€·π‘₯0ξ€Έ,π‘‡βˆ—(π‘₯)ξ€Έξ€Έξ€»βˆ©π‘‡βˆ—ξ€·π‘₯0ξ€Έ.(6.6) Since π‘‡βˆ—(π‘₯0)βˆˆβ„°π‘ž(𝐻), by definition of external π‘ž-hyperconvexity, π½β‰ βˆ… if for each π‘₯∈𝐷, we have dist(𝑇(π‘₯),π‘‡βˆ—(π‘₯0))≀𝑑𝐻(π‘‡βˆ—(π‘₯),π‘‡βˆ—(π‘₯0)) and 𝑇distβˆ—ξ€·π‘₯0ξ€Έξ€Έ,𝑇(π‘₯)β‰€π‘‘π»ξ€·π‘‡βˆ—ξ€·π‘₯0ξ€Έ,π‘‡βˆ—ξ€Έ,(π‘₯)(6.7) and for any π‘₯,π‘¦βˆˆπ·, also 𝑑(𝑇(π‘₯),𝑇(𝑦))β‰€π‘‘π»ξ€·π‘‡βˆ—(π‘₯),π‘‡βˆ—ξ€·π‘₯0ξ€Έξ€Έ+π‘‘π»ξ€·π‘‡βˆ—ξ€·π‘₯0ξ€Έ,π‘‡βˆ—ξ€Έ.(𝑦)(6.8) We are going to check that these conditions hold.
Let π‘₯∈𝐷. For each πœ–>0, we have π‘‡βˆ—(π‘₯)βŠ†π΅π‘‘βˆ’1(π‘‡βˆ—(π‘₯0),𝑑𝐻(π‘‡βˆ—(π‘₯),π‘‡βˆ—(π‘₯0))+πœ–) and π‘‡βˆ—(π‘₯)βŠ†π΅π‘‘(π‘‡βˆ—(π‘₯0),𝑑𝐻(π‘‡βˆ—(π‘₯0),π‘‡βˆ—(π‘₯))+πœ–) by definition of the Hausdorff quasipseudometric.
Since 𝑇(π‘₯)βˆˆπ‘‡βˆ—(π‘₯), for each πœ–>0, there is π‘Žβˆˆπ‘‡βˆ—(π‘₯0) such that 𝑑(𝑇(π‘₯),π‘Ž)≀𝑑𝐻(π‘‡βˆ—(π‘₯),π‘‡βˆ—(π‘₯0))+πœ–, and there is π‘βˆˆπ‘‡βˆ—(π‘₯0) such that 𝑑(𝑏,𝑇(π‘₯))β‰€π‘‘π»ξ€·π‘‡βˆ—ξ€·π‘₯0ξ€Έ,π‘‡βˆ—ξ€Έ(π‘₯)+πœ–.(6.9) Therefore, dist(𝑇(π‘₯),π‘‡βˆ—(π‘₯0))≀𝑑𝐻(π‘‡βˆ—(π‘₯),π‘‡βˆ—(π‘₯0)) and dist(π‘‡βˆ—(π‘₯0),𝑇(π‘₯))≀𝑑𝐻(π‘‡βˆ—(π‘₯0),π‘‡βˆ—(π‘₯)).
We finally also note that by assumption on 𝑇, for each π‘₯,π‘¦βˆˆπ· we have that 𝑑(𝑇(π‘₯),𝑇(𝑦))β‰€π‘‘π»ξ€·π‘‡βˆ—(π‘₯),π‘‡βˆ—ξ€Έ(𝑦)β‰€π‘‘π»ξ€·π‘‡βˆ—(π‘₯),π‘‡βˆ—ξ€·π‘₯0ξ€Έξ€Έ+π‘‘π»ξ€·π‘‡βˆ—ξ€·π‘₯0ξ€Έ,π‘‡βˆ—ξ€Έ.(𝑦)(6.10) Thus, we have shown that π½β‰ βˆ…. Choose 𝑦0∈𝐽 and define
𝑇(π‘₯)=𝑦0 if π‘₯=π‘₯0 and 𝑇(π‘₯)=𝑇(π‘₯) if π‘₯∈𝐷.
Since for each π‘₯∈𝐷,   𝑑(𝑇(π‘₯0),𝑇(π‘₯))=𝑑(𝑦0,𝑇(π‘₯))≀𝑑𝐻(π‘‡βˆ—(π‘₯0),π‘‡βˆ—(π‘₯)) and 𝑑(𝑇(π‘₯),𝑇(π‘₯0))=𝑑(𝑇(π‘₯),𝑦0)≀𝑑𝐻(π‘‡βˆ—(π‘₯),π‘‡βˆ—(π‘₯0)), we conclude that (𝐷βˆͺ{π‘₯0},𝑇)βˆˆβ„±, contradicting the maximality of (𝐷,𝑇). Therefore, 𝐷=𝑋.

Corollary 6.7 (compare [18, Corollary 1]). Let (𝐻,𝑑) be a π‘ž-hyperconvex 𝑇0-quasimetric space. Moreover, let (𝑋,𝜌) be a 𝑇0-quasimetric space, and suppose that π‘‡βˆ—βˆΆπ‘‹β†’β„°π‘ž(𝐻) is nonexpansive, that is, 𝑑𝐻(π‘‡βˆ—(π‘₯),π‘‡βˆ—(𝑦))β‰€πœŒ(π‘₯,𝑦) whenever π‘₯,π‘¦βˆˆπ‘‹. Then, there is a nonexpansive map π‘‡βˆΆ(𝑋,𝜌)β†’(𝐻,𝑑) for which 𝑇(π‘₯)βˆˆπ‘‡βˆ—(π‘₯) whenever π‘₯βˆˆπ‘‹.

Proof. Because π‘‡βˆ— is nonexpansive, the selection obtained from Theorem 6.6 is also nonexpansive.

Corollary 6.8 (compare [18, Corollary 2]). Let 𝐻 be a bounded and π‘ž-hyperconvex 𝑇0-quasimetric space and suppose that π‘‡βˆ—βˆΆπ»β†’β„°π‘ž(𝐻) is nonexpansive. Then π‘‡βˆ— has a fixed point, that is, there exists π‘₯∈𝐻 such that π‘₯βˆˆπ‘‡βˆ—(π‘₯).

Proof. The existence of a fixed point for the nonexpansive selection 𝑇 of π‘‡βˆ—, which exists by Corollary 6.7, follows from Theorem 3.3.

In the following theorem, we set Fix(π‘‡βˆ—)={π‘₯∈𝐻∢π‘₯βˆˆπ‘‡βˆ—(π‘₯)}. According to Corollary 6.8,  Fix(π‘‡βˆ—)β‰ βˆ… if 𝐻 is bounded and π‘ž-hyperconvex, and π‘‡βˆ— is nonexpansive.

Theorem 6.9 (compare [18, Theorem 2]). Let (𝐻,𝑑) be a π‘ž-hyperconvex 𝑇0-quasimetric space, let π‘‡βˆ—βˆΆπ»β†’β„°π‘ž(𝐻) be a nonexpansive map and suppose that Fix(π‘‡βˆ—)β‰ βˆ…. Then, there exists a nonexpansive map π‘‡βˆΆπ»β†’π» with 𝑇(π‘₯)βˆˆπ‘‡βˆ—(π‘₯) whenever π‘₯∈𝐻 and for which Fix(𝑇)=Fix(π‘‡βˆ—).

Proof. Let β„± denote the collection of all pairs (𝐷,𝑇), where Fix(π‘‡βˆ—)βŠ†π·βŠ†π»,π‘‡βˆΆπ·β†’π»,𝑇(𝑑)βˆˆπ‘‡βˆ—(𝑑) whenever π‘‘βˆˆπ·,𝑇(π‘₯)=π‘₯ whenever π‘₯∈Fix(π‘‡βˆ—), and 𝑑(𝑇(π‘₯),𝑇(𝑦))≀𝑑(π‘₯,𝑦) whenever π‘₯,π‘¦βˆˆπ·. By assumption (Fix(π‘‡βˆ—),Id)βˆˆβ„±, so β„±β‰ βˆ…. The argument now follows from a modification of the proof of Theorem 6.6. One defines a partial order on β„± by setting (𝐷1,𝑇1)β‰Ό(𝐷2,𝑇2) if and only if 𝐷1βŠ†π·2 and 𝑇2|𝐷1=𝑇1.
Let ((𝐷𝛼,𝑇𝛼))π›Όβˆˆπ‘† be an increasing chain in (β„±,β‰Ό). Then, it follows that (β‹ƒπ›Όβˆˆπ‘†π·π›Ό,𝑇)βˆˆβ„± where 𝑇|𝐷𝛼=𝑇𝛼 whenever π›Όβˆˆπ‘†. By Zorn's lemma, (β„±,β‰Ό) has a maximal element, say (𝐷,𝑇). Assume 𝐷≠𝐻 and find π‘₯0∈𝐻⧡𝐷. Set 𝐷=𝐷βˆͺ{π‘₯0} and consider the set: 𝐽=π‘₯βˆˆπ·ξ€ΊπΆπ‘‘ξ€·π‘‡ξ€·(π‘₯),𝑑π‘₯,π‘₯0ξ€Έξ€Έβˆ©πΆπ‘‘βˆ’1𝑇π‘₯(π‘₯),𝑑0,π‘₯ξ€Έξ€Έξ€»βˆ©π‘‡βˆ—ξ€·π‘₯0ξ€Έ.(6.11) Since π‘‡βˆ—(π‘₯0)βˆˆβ„°π‘ž(𝐻), by definition of external π‘ž-hyperconvexity, π½β‰ βˆ… if for each π‘₯∈𝐷, we have dist(𝑇(π‘₯),π‘‡βˆ—(π‘₯0))≀𝑑(π‘₯,π‘₯0) and dist(π‘‡βˆ—(π‘₯0),𝑇(π‘₯))≀𝑑(π‘₯0,π‘₯), and for any π‘₯,π‘¦βˆˆπ· we have 𝑑(𝑇(π‘₯),𝑇(𝑦))≀𝑑(π‘₯,π‘₯0)+𝑑(π‘₯0,𝑦).
We are going to check these conditions next. Let π‘₯∈𝐷. For each πœ–>0, we have π‘‡βˆ—(π‘₯)βŠ†π΅π‘‘βˆ’1(π‘‡βˆ—(π‘₯0),𝑑𝐻(π‘‡βˆ—(π‘₯),π‘‡βˆ—(π‘₯0))+πœ–)βŠ†π΅π‘‘βˆ’1(π‘‡βˆ—(π‘₯0),𝑑(π‘₯,π‘₯0)+πœ–) and π‘‡βˆ—(π‘₯)βŠ†π΅π‘‘(π‘‡βˆ—(π‘₯0),𝑑𝐻(π‘‡βˆ—(π‘₯0),π‘‡βˆ—(π‘₯))+πœ–)βŠ†π΅π‘‘(π‘‡βˆ—(π‘₯0),𝑑(π‘₯0,π‘₯)+πœ–) by definition of the Hausdorff quasipseudometric.
Since 𝑇(π‘₯)βˆˆπ‘‡βˆ—(π‘₯), for each πœ–>0, there is π‘Žβˆˆπ‘‡βˆ—(π‘₯0) such that 𝑑(𝑇(π‘₯),π‘Ž)≀𝑑(π‘₯,π‘₯0)+πœ–, and there is π‘βˆˆπ‘‡βˆ—(π‘₯0) such that 𝑑(𝑏,𝑇(π‘₯))≀𝑑(π‘₯0,π‘₯)+πœ–. Therefore, dist(𝑇(π‘₯),π‘‡βˆ—(π‘₯0))≀𝑑(π‘₯,π‘₯0) and dist(π‘‡βˆ—(π‘₯0),𝑇(π‘₯))≀𝑑(π‘₯0,π‘₯).
We finally also note that by assumption on 𝑇 for each π‘₯,π‘¦βˆˆπ· we have that 𝑑(𝑇(π‘₯),𝑇(𝑦))≀𝑑(π‘₯,𝑦)≀𝑑(π‘₯,π‘₯0)+𝑑(π‘₯0,𝑦).
Thus, we have shown that π½β‰ βˆ…. Choose 𝑦0∈𝐽 and define 𝑇(π‘₯)=𝑦0 if π‘₯=π‘₯0 and 𝑇(π‘₯)=